Dimensions of automorphism group schemes of finite level truncations of $F$-cyclic $F$-crystals

Abstract

Let $\mathcal{M}_{\pi}$ be an $F$-cyclic $F$-crystal $\mathcal{M}_{\pi}$ over an algebraically closed field defined by a permutation $\pi$ and a set of prescribed Hodge slopes. We prove combinatorial formulas for the dimension $\gamma_{\mathcal{M}_{\pi}}(m)$ of the automorphism group scheme of $\mathcal{M}_{\pi}$ at finite level $m$ and the number of connected components of the endomorphism group scheme of $\mathcal{M}_{\pi}$ at finite level $m$. As an application, we show that if $\mathcal{M}_{\pi}$ is a nonordinary Dieudonn\'e module defined by a cycle $\pi$, then $\gamma_{\mathcal{M}_{\pi}}(m+1) - \gamma_{\mathcal{M}_{\pi}}(m) < \gamma_{\mathcal{M}_{\pi}}(m) - \gamma_{\mathcal{M}_{\pi}}(m-1)$ for all $1 \leq m \leq n_{\mathcal{M}_{\pi}}$, where $n_{\mathcal{M}_{\pi}}$ is the isomorphism number of $\mathcal{M}_{\pi}$.